Cryptographic method of multilayer diffusion in multidimension

ABSTRACT

The invention provides a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), in which a multidimensional medium is meanwhile overlapped to the diffusion-area; accordingly, repeating the diffusion function for at least one time thus brings about the multilayer effect. FIG.  1  shows an embodiment of the present invention in flow chart diagram form, comprising of: inputting a plaintext in encryption or a ciphertext in decryption  100 ; inputting a series of password data forward in encryption or backward in decryption  200 ; further, by the password data, converting the dimensions of the plaintext  300 , and implementing with a diffusion function, repeated T E  times in encryption, T D  times in decryption  400 ; outputting the ciphertext in encryption or the plaintext in decryption  600  if completing all password data  500.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a cryptographic method. More particularly, the invention relates to a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), and further, repeating the diffusion function for at least one time to create a multilayer effect in order to perform the encryption and the decryption.

2. Description of the Related Art

The Applicant's following patent applications are related to the invention and are incorporated herein by reference: “Diffused Data Encryption/Decryption Processing Method”, application Ser. No. 12/365,160, filed Feb. 3, 2009 (CIP of application Ser. No. 10/963,014, filed Oct. 12, 2004); “Multipoint Synchronous Diffused Encryption/Decryption Method”, application Ser. No. 11/171,549, filed Jun. 30, 2005.

The prior art described that the coding of a 2D diffusion-area, see application Ser. No. 12/365,160, page 7, teaches the math of A(i, j)=A⊕Ac_(i)⊕Ar_(j)⊕b(i, j); further expressed the status of diffusion from inward to outward or vice versa in reverse, and implemented to multidimensional matrix A(i₁, i₂, . . . i_(n)), see application Ser. No. 11/171,549, page 4, 7.

The present invention emphasizes the multilayer effect of multidimensional diffusion. The diffusion function herein is notated specially by AF(p₁, p₂, . . . p_(n)) to differentiate from the traditional symbolization of matrix position.

SUMMARY OF THE INVENTION

The invention provides a diffusion function working on a multidimensional diffusion-area (plaintext/ciphertext), in which a multidimensional medium is meanwhile overlapped to the diffusion-area; accordingly, repeating the diffusion function for at least one time thus brings about the multilayer effect. In addition, the numbers of repetition, a so called diffusion-cycle, is then able to divide into two parts: one for encrypting, the other for decrypting; consequently, the original status of the diffusion-area is recovered through the diffusion-cycle. The steps are shown as follows:

-   -   (a) Selecting a diffusion function, a multidimensional medium;     -   (b) Inputting a multidimensional diffusion-area         (plaintext/ciphertext), for which the dimensions are the same as         the medium's, and generating a diffusion-cycle;     -   (c) Repeating the diffusion function working on a plaintext for         a first part of the diffusion-cycle to generate a ciphertext;     -   (d) Repeating the diffusion function working on the ciphertext         for a second part of the diffusion-cycle to recover the         plaintext.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a summary flow chart diagram showing the main steps taken while encrypting/decrypting by repeating diffusion function in accordance with the present invention;

FIG. 2A is a summary flow chart diagram of FIG. 1, 410 showing the steps taken while originating the function of point-diffusion from a diffusion-center in accordance with the present invention;

FIG. 2B is a two-dimension visualized diagram of FIG. 2A showing the point-diffusion by way of a medium anchoring to a diffusion-center in accordance with the present invention;

FIG. 3A is a summarized flow chart diagram of FIG. 1, 420 showing the steps taken while originating the function of block-diffusion from a block diffusion-center in accordance with the present invention;

FIG. 3B is a two-dimension visualized diagram of FIG. 3A showing the block-diffusion by way of a medium anchoring to a diffusion-center, a block anchoring to the diffusion-center to form a block diffusion-center in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows an embodiment of the present invention in flow chart diagram form. This system comprises of: inputting a plaintext in encryption or a ciphertext in decryption 100; inputting a series of password data forward in encryption or backward in decryption 200; further, by the password data, converting the dimensions of the plaintext 300, and implementing with a function of diffusion, repeated T_(E) times in encryption, T_(D) times in decryption 400; outputting the ciphertext in encryption or the plaintext in decryption 600 if completing all password data 500.

Function of Point-Diffusion:

FIG. 2A shows an embodiment of the point-diffusion function, FIG. 1, 410, in flow chart diagram. The function comprises of: reading a diffusion-area (plaintext/ciphertext), a diffusion-center, and a medium with an anchor-point 201; anchoring the medium to the diffusion-center with the anchor-point 411; implementing the point-diffusion AF(p₁, p₂, . . . p_(n)) 412, which is further detailed in Notation of Point-Diffusion. In addition, also see FIG. 2B with 2D visualization for a more clear view, the diffusion effect colored from white to black generates the column segments a-g, a-b for later diffusion calculation.

Notation of Point-Diffusion:

-   A: a diffusion-area, wherein A expresses a d₁×d₂× . . . ×d_(n)     binary matrix, wherein A includes a diffusion-center {dot over (P)}     expressed (p₁, p₂, . . . p_(n)) coordinate position. -   S: a n-dimension medium, expresses a s₁×s₂× . . . ×s_(n) binary     matrix, wherein S includes an anchor-point {dot over (S)} expressed     (s₁, s₂, . . . , s_(n)) coordinate position. -   AF(p₁, p₂, . . . p_(n)): the diffusion-area A performs the function     of point-diffusion at position {dot over (P)}, wherein S overlaps A     by {dot over (S)} anchoring to {dot over (P)}; further comprising:     -   AF(p₁, p₂, . . . , p_(n))=A⊕Ad_(1p)⊕Ad_(2p)⊕ . . . ⊕Ad_(np)⊕S;     -   Ad_(ip)=[A_(d) _(i) (2), . . . , A_(d) _(i) (p_(i)), A_(d) _(i)         (0), A_(d) _(i) (p_(i)), . . . , A_(d) _(i) (d_(i)−1)];     -   Ad_(ip) expresses a series of n−1 dimensional binary matrix         A_(d) _(i) on the axis d_(i). Furthermore, A_(d) _(i) (p_(i))         represents the original A_(d) _(i) the coordinate p_(i), and         then, A_(d) _(i) (0) expresses a zero matrix filling at the         coordinate p_(i).     -   For example: 2D point-diffusion, with rows for x, columns for y,         AF(p_(x)=3, p_(y)=2).         -   Suppose

${A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}},{S = \begin{bmatrix} s_{11} & s_{12} & s_{13} & s_{14} \\ s_{21} & s_{22} & s_{23} & s_{24} \\ s_{31} & s_{32} & s_{33} & s_{34} \\ s_{41} & s_{42} & s_{43} & s_{44} \end{bmatrix}},{\overset{.}{S} = \left( {2,1} \right)}$ thus $\begin{matrix} {{{AF}\left( {3,2} \right)} = {A \oplus {Ax}_{3} \oplus {Ay}_{2} \oplus S}} \\ {= {A \oplus \begin{bmatrix} a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ 0 & 0 & 0 & 0 \\ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix} \oplus \begin{bmatrix} a_{12} & 0 & a_{12} & a_{13} \\ a_{22} & 0 & a_{22} & a_{23} \\ a_{32} & 0 & a_{32} & a_{33} \\ a_{42} & 0 & a_{42} & a_{43} \end{bmatrix} \oplus}} \\ {\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & s_{11} & s_{12} & s_{13} \\ 0 & s_{21} & s_{22} & s_{23} \\ 0 & s_{31} & s_{32} & s_{33} \end{bmatrix}} \end{matrix}$

In detail, Ax₃ expresses a series of one dimensional binary matrixes A_(x) on the axis x; wherein Ax₃ comprises A_(x) (2)=[a₂₁ a₂₂ a₂₃ a₂₄] to position 1, A_(x) (3)=[a₃₁ a₃₂ a₃₃ a₃₄] to positions 2, 4, and A_(x) (0)=[0 0 0 0] at position 3. Furthermore, Ay₂ expresses a series of one dimensional binary matrixes A_(y) on the axis y; wherein Ay₂ comprises

${A_{y}(2)} = \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \\ a_{42} \end{bmatrix}$ to positions 1, 3,

${A_{y}(3)} = \begin{bmatrix} a_{13} \\ a_{23} \\ a_{33} \\ a_{43} \end{bmatrix}$ to position 4, and

${A_{y}(0)} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ at position 2.

-   -   Finally, the effective S comes from the overlap between S and A,         while {dot over (S)}=(2,1) anchors to P=(3,2).     -   For example: 3D point-diffusion AF(p_(x)=3, p_(y)=2, p_(z)=1).         Suppose

${A = \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} a_{111} & a_{121} & a_{131} & a_{141} \\ a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ a_{411} & a_{421} & a_{431} & a_{441} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{112} & a_{122} & a_{132} & a_{142} \\ a_{212} & a_{222} & a_{232} & a_{242} \\ a_{312} & a_{322} & a_{332} & a_{342} \\ a_{412} & a_{422} & a_{432} & a_{442} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{113} & a_{123} & a_{133} & a_{143} \\ a_{213} & a_{223} & a_{233} & a_{243} \\ a_{313} & a_{323} & a_{333} & a_{343} \\ a_{413} & a_{423} & a_{433} & a_{443} \end{matrix}}}} \right\rbrack},{S = \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} s_{111} & s_{121} & s_{131} & s_{141} \\ s_{211} & s_{221} & s_{231} & s_{241} \\ s_{311} & s_{321} & s_{331} & s_{341} \\ s_{411} & s_{421} & s_{431} & s_{441} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} s_{112} & s_{122} & s_{132} & s_{142} \\ s_{212} & s_{222} & s_{232} & s_{242} \\ s_{312} & s_{322} & s_{332} & s_{342} \\ s_{412} & s_{422} & s_{432} & s_{442} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} s_{113} & s_{123} & s_{133} & s_{143} \\ s_{213} & s_{223} & s_{233} & s_{243} \\ s_{313} & s_{323} & s_{333} & s_{343} \\ s_{413} & s_{423} & s_{433} & s_{443} \end{matrix}}}} \right\rbrack},{{\overset{.}{S} = \left( {2,1,3} \right)};{thus}},{{{AF}\left( {3,2,1} \right)} = {{A \oplus {Ax}_{3} \oplus {Ay}_{2} \oplus {Az}_{1} \oplus S} = {{A\mspace{14mu}\ldots}\mspace{14mu} \oplus \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ 0 & 0 & 0 & 0 \\ a_{311} & a_{321} & a_{331} & a_{341} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{212} & a_{222} & a_{232} & a_{242} \\ a_{312} & a_{322} & a_{332} & a_{342} \\ 0 & 0 & 0 & 0 \\ a_{312} & a_{322} & a_{332} & a_{342} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{213} & a_{223} & a_{233} & a_{243} \\ a_{313} & a_{323} & a_{333} & a_{343} \\ 0 & 0 & 0 & 0 \\ a_{313} & a_{323} & a_{333} & a_{343} \end{matrix}}}} \right\rbrack \oplus \mspace{281mu}\left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} a_{121} & 0 & a_{121} & a_{131} \\ a_{221} & 0 & a_{221} & a_{231} \\ a_{321} & 0 & a_{321} & a_{331} \\ a_{421} & 0 & a_{421} & a_{431} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{122} & 0 & a_{122} & a_{132} \\ a_{222} & 0 & a_{222} & a_{232} \\ a_{322} & 0 & a_{322} & a_{332} \\ a_{422} & 0 & a_{422} & a_{432} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{123} & 0 & a_{123} & a_{133} \\ a_{223} & 0 & a_{223} & a_{233} \\ a_{323} & 0 & a_{323} & a_{333} \\ a_{423} & 0 & a_{423} & a_{433} \end{matrix}}}} \right\rbrack \oplus \mspace{315mu}\left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{111} & a_{121} & a_{131} & a_{141} \\ a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ a_{411} & a_{421} & a_{431} & a_{441} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{112} & a_{122} & a_{132} & a_{142} \\ a_{212} & a_{222} & a_{232} & a_{242} \\ a_{312} & a_{322} & a_{332} & a_{342} \\ a_{412} & a_{422} & a_{432} & a_{442} \end{matrix}}}} \right\rbrack \oplus \mspace{509mu}{\left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & s_{113} & s_{123} & s_{133} \\ 0 & s_{213} & s_{223} & s_{233} \\ 0 & s_{313} & s_{323} & s_{333} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}}}} \right\rbrack.}}}}$

In detail, Ax₃ expresses a series of two dimensional binary matrixes A_(x) on the axis x; wherein Ax₃ comprises

${A_{x}(2)} = \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} a_{211} & a_{221} & a_{231} \end{matrix}a_{241}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{212} & a_{222} & a_{232} & a_{242} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{213} & a_{223} & a_{233} & a_{243} \end{matrix}}}} \right\rbrack$ to position 1.

${A_{x}(3)} = \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} a_{311} & a_{321} & a_{331} \end{matrix}a_{341}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} a_{312} & a_{322} & a_{332} & a_{342} \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} a_{313} & a_{323} & a_{333} & a_{343} \end{matrix}}}} \right\rbrack$ to positions 2, 4, and

${A_{x}(0)} = \left\lbrack {\overset{z = 1}{\overset{︷}{\begin{matrix} 0 & 0 & \begin{matrix} 0 & 0 \end{matrix} \end{matrix}}}{\overset{z = 2}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}}}}\overset{z = 3}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}}}} \right\rbrack$ at position 3. Furthermore, Ay₂ expresses a series of two dimensional binary matrixes A_(y) on the axis y; wherein Ay₂ comprises

${A_{y}(2)} = \begin{bmatrix} \overset{\overset{z = 1}{︷}}{a_{121}} & \overset{\overset{z = 2}{︷}}{a_{122}} & \overset{\overset{z = 3}{︷}}{a_{123}} \\ a_{221} & a_{222} & a_{223} \\ a_{321} & a_{322} & a_{323} \\ a_{421} & a_{422} & a_{423} \end{bmatrix}$ to positions 1, 3,

${A_{y}(3)} = \begin{bmatrix} \overset{\overset{z = 1}{︷}}{a_{131}} & \overset{\overset{z = 2}{︷}}{a_{132}} & \overset{\overset{z = 3}{︷}}{a_{133}} \\ a_{231} & a_{232} & a_{233} \\ a_{331} & a_{332} & a_{333} \\ a_{431} & a_{432} & a_{433} \end{bmatrix}$ to position 4, and

${A_{y}(0)} = \begin{bmatrix} \overset{\overset{z = 1}{︷}}{0} & \overset{\overset{z = 2}{︷}}{0} & \overset{\overset{z = 3}{︷}}{0} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ at position 2. Moreover, Az₁ expresses a series of two dimensional binary matrixes A_(z) on the axis z; wherein Az₁ comprises

${A_{z}(1)} = \begin{bmatrix} a_{111} & a_{121} & a_{131} & a_{141} \\ a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ a_{411} & a_{421} & a_{431} & a_{441} \end{bmatrix}$ to position 2,

${A_{z}(2)} = \begin{bmatrix} a_{112} & a_{122} & a_{132} & a_{142} \\ a_{212} & a_{222} & a_{232} & a_{242} \\ a_{312} & a_{322} & a_{332} & a_{342} \\ a_{412} & a_{422} & a_{432} & a_{442} \end{bmatrix}$ to position 3, and

${A_{z}(0)} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$ at position 1. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1,3) anchors to P=(3,2,1).

-   AF(p₁, p₂ ^(t), . . . , p_(n)): A performs the function of     point-diffusion, repeated t times.     -   Example: (a) AF(p₁, p₂ ², . . . , p_(n))=AF(p₁, p₂, . . . ,         p_(n))F(p₁, p₂, . . . , p_(n))         -   (b) AF(p₁, p₂ ¹, . . . , p_(n))=AF(p₁, p₂, . . . , p_(n))         -   (c) AF(p₁, p₂ ⁰, . . . , p_(n))=A -   T: a diffusion-cycle, expresses AF(p₁, p₂ ^(T), . . . , p_(n))=A,     wherein T=2^(U+1), U=┌log₂ u┐, u=max(d₁, d₂, . . . , d_(n)).     Function of Block-Diffusion:

FIG. 3A shows an embodiment of the block-diffusion function, FIG. 1, 420, in flow chart diagram. The function comprises of: reading a diffusion-area (plaintext/ciphertext), a diffusion-center, a medium with an anchor-point and a block with an anchor-point 202; anchoring the medium and the block to the diffusion-center with the anchor-point 421; implementing the block-diffusion ÂF({circumflex over (p)}₁, {circumflex over (p)}₂, . . . {circumflex over (p)}_(n)) 422, further detailed in Notation of Block-Diffusion. In addition, also see FIG. 3B with 2D visualization for a more clear view, the diffusion effect colored from white to black generates the block-column segments a-c, a-b for later diffusion calculation.

Notation of Block-Diffusion:

-   A: a n-dimension plaintext, expresses a d₁×d₂× . . . ×d_(n) binary     matrix, wherein A includes a diffusion-center P expressed (p₁, p₂, .     . . p_(n)) coordinate position. -   S: a n-dimension medium, expresses a s₁×s₂× . . . ×s_(n) binary     matrix, wherein S includes an anchor-point {dot over (S)} expressed     ({dot over (s)}₁, {dot over (s)}₂, . . . , {dot over (s)}_(n))     coordinate position. -   B: a n-dimension unit-block, expresses a b₁×b₂× . . . ×b_(n) binary     matrix, wherein B includes an anchor-point {dot over (B)} expressed     ({dot over (b)}₁, {dot over (b)}₂, . . . , {dot over (b)}_(n))     coordinate position. -   ÂF({circumflex over (p)}₁, {circumflex over (p)}₂, . . . {circumflex     over (p)}_(n)): Â performs the function of block-diffusion, wherein     Â expresses A by B unit seeing that {dot over (B)} anchors to P, and     thus, includes a block diffusion-center {circumflex over (P)}     expressed ({circumflex over (p)}₁, {circumflex over (p)}₂, . . .     {circumflex over (p)}_(n)) coordinate position. Therefore, A     translates into a {circumflex over (d)}₁×{circumflex over (d)}₂× . .     . ×{circumflex over (d)}_(n) binary matrix, wherein {circumflex over     (d)}_(i)=┌(p_(i)−{dot over (b)}_(i))/b_(i)┐+┌(d_(i)−p_(i)+{dot over     (b)}_(i)/b_(i)┐, and {circumflex over (p)}_(i)=|(p_(i)−{dot over     (b)}_(i))/b_(i)|+1; further comprising:

ÂF(p̂₁, p̂₂, …  , p̂_(n)) = Â ⊕ Âd̂_(1p̂) ⊕ Âd̂_(2p̂) ⊕ … ⊕ Âd̂_(np̂) ⊕ S; Âd̂_(ip̂) = [Â_(d̂_(i))(2), …  , Â_(d̂_(i))(p̂_(i)), Â_(d̂_(i))(0), Â_(d̂_(i))(p̂_(i)), …  , Â_(d̂_(i))(d̂_(i) − 1)]; Â{circumflex over (d)}_(i{circumflex over (p)}) expresses a series of n−1 dimensional binary matrixes

Â_(d̂_(i)) on the axis {circumflex over (d)}_(i). Furthermore,

Â_(d̂_(i))(p̂_(i)) represents the original

Â_(d̂_(i)) at the coordinate {circumflex over (p)}_(i), and then,

Â_(d̂_(i))(0) expresses a zero matrix tilling at the coordinate {circumflex over (p)}_(i).

-   -   For example: 2D block-diffusion, with rows for x, columns for y,         AF(p_(x)=3, p_(y)=2). Suppose

${A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}},{S = \begin{bmatrix} s_{11} & s_{12} & s_{13} & s_{14} \\ s_{21} & s_{22} & s_{23} & s_{24} \\ s_{31} & s_{32} & s_{33} & s_{34} \\ s_{41} & s_{42} & s_{43} & s_{44} \end{bmatrix}},{B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}},{\overset{.}{S} = \left( {2,1} \right)},{{\overset{.}{B} = \left( {1,1} \right)};}$ thus, dimensions {circumflex over (x)}=┌(3−1)/2┐+┌(4−3+1)/2┐=2 and ŷ=┌(2−1)/2┐+┌(4−2+1)/2┐=3;

that shows the block-diffusion in 2×3 blocks, but with the data still kept in 4×4 bits. And now {circumflex over (p)}_(x)=┌(3−1)/2┐+1=2, {circumflex over (p)}_(y)=┌(2−1)/2┐+1=2, thus

In detail, Â{circumflex over (x)}₂ expresses a series of one dimensional binary matrixes Â_({circumflex over (x)}) on the axis {circumflex over (x)}; wherein Â{circumflex over (x)}₂ comprises

${{\hat{A}}_{\hat{x}}(2)} = \begin{bmatrix} a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}$ to position 1, and

${{\hat{A}}_{\hat{x}}(0)} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$ at position 2. Furthermore, Âŷ₂ expresses a series of one dimensional binary matrixes Â_(ŷ) on the axis ŷ; wherein Âŷ₂ comprises

${{\hat{A}}_{\hat{y}}(2)} = \begin{bmatrix} a_{12} & a_{13} \\ a_{22} & a_{23} \\ a_{32} & a_{33} \\ a_{42} & a_{43} \end{bmatrix}$ to positions 1, 3, and

${{\hat{A}}_{\hat{y}}(0)} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$ at position 2. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1) anchors to P=(3,2).

-   -   For example: 3D block-diffusion AF(p_(x)=3, p_(y)=2, p_(z)=1).         Suppose

${A = \left\lbrack {\overset{\overset{z = 1}{︷}}{\begin{matrix} a_{111} & a_{121} & a_{131} & a_{141} \\ a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ a_{411} & a_{421} & a_{431} & a_{441} \end{matrix}}❘{\overset{\overset{z = 2}{︷}}{\begin{matrix} a_{112} & a_{122} & a_{132} & a_{142} \\ a_{212} & a_{222} & a_{232} & a_{242} \\ a_{312} & a_{322} & a_{332} & a_{342} \\ a_{412} & a_{422} & a_{432} & a_{442} \end{matrix}}❘\overset{\overset{z = 3}{︷}}{\begin{matrix} a_{113} & a_{123} & a_{133} & a_{143} \\ a_{213} & a_{223} & a_{233} & a_{243} \\ a_{313} & a_{323} & a_{333} & a_{343} \\ a_{413} & a_{423} & a_{433} & a_{443} \end{matrix}}}} \right\rbrack},{S = \left\lbrack {\overset{\overset{z = 1}{︷}}{\begin{matrix} s_{111} & s_{121} & s_{131} & s_{141} \\ s_{211} & s_{221} & s_{231} & s_{241} \\ s_{311} & s_{321} & s_{331} & s_{341} \\ s_{411} & s_{421} & s_{431} & s_{441} \end{matrix}}❘{\overset{\overset{z = 2}{︷}}{\begin{matrix} s_{112} & s_{122} & s_{132} & s_{142} \\ s_{212} & s_{222} & s_{232} & s_{242} \\ s_{312} & s_{322} & s_{332} & s_{342} \\ s_{412} & s_{422} & s_{432} & s_{442} \end{matrix}}❘\overset{\overset{z = 3}{︷}}{\begin{matrix} s_{113} & s_{123} & s_{133} & s_{143} \\ s_{213} & s_{223} & s_{233} & s_{243} \\ s_{313} & s_{323} & s_{333} & s_{343} \\ s_{413} & s_{423} & s_{433} & s_{443} \end{matrix}}}} \right\rbrack},\text{}{B = \left\lbrack \overset{\overset{z = 1}{︷}}{\begin{matrix} b_{111} & b_{121} \\ b_{211} & b_{221} \end{matrix}} \middle| \overset{\overset{z = 2}{︷}}{\begin{matrix} b_{112} & b_{122} \\ b_{212} & b_{222} \end{matrix}} \right\rbrack},{\overset{.}{S} = \left( {2,1,3} \right)},{{\overset{.}{B} = \left( {1,1,1} \right)};}$ thus, dimensions {circumflex over (x)}=┌(3−1)/2┐+┌(4−3+1)/2┐=2, ŷ=┌(2−1)/2┐+┌(4−2+1)/2┐=3, and {circumflex over (z)}=┌(1−1)/2┐+┌(4−1+1)/2┐=2; further

-   -   that shows the block-diffusion in 2×3×2 blocks, but with the         data still kept in 4×4×3 bits. And now {circumflex over         (p)}_(x)=┌(3−1)/2┐+1=2, {circumflex over (p)}_(y)=┌(2−1)/2┐+1=2,         {circumflex over (p)}_(z)=┌(1−1)/2┐+1=1, thus,

In detail, Â{circumflex over (x)}₂ expresses a series of two dimensional binary matrixes Â_({circumflex over (x)}) on the axis {circumflex over (x)}; wherein Â{circumflex over (x)}₂ comprises

${{\hat{A}}_{\hat{x}}(2)} = \left\lbrack {\overset{\hat{z} = 1}{\overset{︷}{\begin{matrix} a_{311} & a_{321} & a_{331} & a_{341} & a_{312} & a_{322} & a_{332} & a_{342} \\ a_{411} & a_{421} & a_{431} & a_{441} & a_{412} & a_{422} & a_{432} & a_{442} \end{matrix}}}\overset{\hat{z} = 2}{\overset{︷}{\begin{matrix} a_{313} & a_{323} & a_{333} & a_{343} \\ a_{413} & a_{423} & a_{433} & a_{443} \end{matrix}}}} \right\rbrack$ to position 1, and

${{\hat{A}}_{\hat{x}}(0)} = \left\lbrack {\overset{\hat{z} = 1}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}}}\overset{\hat{z} = 2}{\overset{︷}{\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}}}} \right\rbrack$ at position 2. Furthermore, Âŷ₂ expresses a series of two dimensional binary matrixes Â_(ŷ) on the axis ŷ; wherein Âŷ₂ comprises Â_(ŷ)(2) to positions 1, 3 is equal to

${\left\lbrack {\overset{\overset{\hat{z} = 1}{︷}}{\begin{matrix} a_{121} & a_{131} \\ a_{221} & a_{231} \\ a_{321} & a_{331} \\ a_{421} & a_{431} \end{matrix}❘\begin{matrix} a_{122} & a_{132} \\ a_{222} & a_{232} \\ a_{322} & a_{332} \\ a_{422} & a_{432} \end{matrix}}❘{\overset{\overset{\hat{z} = 2}{︷}}{\begin{matrix} a_{123} & a_{133} \\ a_{223} & a_{233} \\ a_{323} & a_{333} \\ a_{423} & a_{433} \end{matrix}}❘}} \right\rbrack,\mspace{14mu}{and}}\;$ ${{\hat{A}}_{\hat{y}}(0)} = {\left\lbrack {\overset{\overset{\hat{z} = 1}{︷}}{\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}❘\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}}❘{\overset{\overset{\hat{z} = 2}{︷}}{\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}}❘}} \right\rbrack{and}}$ at position 2. Moreover, Â{circumflex over (z)}₁ expresses a series of two dimensional binary matrixes Â_({circumflex over (z)}) on the axis {circumflex over (z)}; wherein Â{circumflex over (z)}₁ comprises

${{\hat{A}}_{\hat{z}}(1)} = \left\lbrack \overset{\hat{z} = 1}{\overset{︷}{\begin{matrix} a_{111} & a_{121} & a_{131} & a_{141} \\ a_{211} & a_{221} & a_{231} & a_{241} \\ a_{311} & a_{321} & a_{331} & a_{341} \\ a_{411} & a_{421} & a_{431} & a_{441} \end{matrix}}} \right\rbrack$ to position

${{\hat{A}}_{\hat{z}}(0)} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$ at position 1. Finally, the effective S comes from the overlap between S and A, while {dot over (S)}=(2,1,3) anchors to P=(3,2,1).

-   ÂF({circumflex over (p)}₁, {circumflex over (p)}₂ ^(t) , . . . ,     {dot over (p)}_(n)): Â performs the function of block-diffusion,     repeated t times.     -   Example: (a) ÂF({circumflex over (p)}₁, {circumflex over (p)}₂         ², . . . , {circumflex over (p)}_(n))=ÂF({circumflex over (p)}₁,         {circumflex over (p)}₂, . . . , {circumflex over         (p)}_(n))F({circumflex over (p)}₁, {circumflex over (p)}₂, . . .         {circumflex over (p)}_(n))         -   (b) ÂF({circumflex over (p)}₁, {circumflex over (p)}₂ ¹, . .             . , {circumflex over (p)}_(n))=ÂF({circumflex over (p)}₁,             {circumflex over (p)}₂, . . . , {circumflex over (p)}_(n))         -   (c) ÂF({circumflex over (p)}₁, {circumflex over (p)}₂ ⁰, . .             . , {circumflex over (p)}_(n))=A -   T: a diffusion-cycle, expresses {dot over (A)}F({circumflex over     (p)}₁, {circumflex over (p)}₂ ^(T), . . . {dot over (p)}_(n))=A,     wherein T=2^(U+1), U=┌log₂ u┌, u=max(┌d_(i)/b_(i)┐, 1≦i≦n).     Embodiment of Cryptographic Method:

To make it easier to understand the content of the present invention, examples in detail are described as follows:

Suppose a plaintext A: “smoother”, its ASCII code is 73 6d 6f 6f 74 68 65 72, the binary format is shown as an 8×8 two-dimensional matrix as in Table 1-1.

TABLE 1-1 ASCII 73 6d 6f 6f 74 68 65 72 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

Suppose a password: “Yourlips”, its ASCII code is 59 6f 75 72 6c 69 70 73. For applying to the plaintext, the ASCII code: first, excludes the last digit 3; second, forms into octal format 26 26 75.65 34 46 61 51 34 07; third, adds 1 to each digit; Table 1-2 shows that the password includes 10 diffusion-centers.

TABLE 1-2 ASCII 26 26 75 65 34 46 61 51 34 07 Row 3 3 8 7 4 5 7 6 4 1 Column 7 7 6 6 5 7 2 2 5 8

EXAMPLE 1 The Function of Point-Diffusion in 2D

Supposes

${S_{5 \times 5} = \begin{bmatrix} 10011 \\ 01101 \\ 10111 \\ 10010 \\ 11101 \end{bmatrix}},{{\overset{.}{S} = \left( {1,1} \right)};}$ reads every diffusion-center in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle T=2³⁺¹=16, if 1 time on encryption, then 15 times on decryption. In math, inputs the plaintext A, then runs A¹, A₁ ¹, . . . A₉ ¹ and outputs A₁, A₂, . . . A₁₀ during encryption; inputs the ciphertext A₁₀, then runs A₁₀ ¹⁵, A₉ ¹⁵, . . . A₁ ¹⁵ and outputs A₉, . . . , A₁, A during decryption. The details on the order 1, 5, 10 are shown as below, A_(d) _(i) (0) marked in boldface. Encryption at the 1^(st) Diffusion-Center (3,7):

$\begin{matrix} {A^{1} = {{AF}\left( {3,7} \right)}} \\ {= {A \oplus {Ax}_{3} \oplus {Ay}_{7} \oplus S}} \\ {= {\begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 10110001 \\ 01111010 \\ 00000000 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \end{bmatrix} \oplus \begin{bmatrix} 11100101 \\ 01100000 \\ 11110101 \\ 11101000 \\ 00010000 \\ 11111101 \\ 11111101 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= {\left\lbrack \begin{matrix} 10100110 \\ 10101011 \\ 10001101 \\ 11100111 \\ 11101111 \\ 10001001 \\ 11111110 \\ 11111111 \end{matrix} \right\rbrack = {A_{1}.}}} \end{matrix}$ Encryption at the 5^(th) Diffusion-Center (4,5):

$\begin{matrix} {A_{4}^{1} = {A_{4}{F\left( {4,5} \right)}}} \\ {= {A_{4} \oplus {A_{4}x_{4}} \oplus {A_{4}y_{5}} \oplus S}} \\ {= {\begin{bmatrix} 11010111 \\ 00010101 \\ 00010001 \\ 01000011 \\ 00001111 \\ 01011001 \\ 10101011 \\ 01100101 \end{bmatrix} \oplus \begin{bmatrix} 00010101 \\ 00010001 \\ 01000011 \\ 00000000 \\ 01000011 \\ 00001111 \\ 01011001 \\ 10101011 \end{bmatrix} \oplus \begin{bmatrix} 10100011 \\ 00100010 \\ 00100000 \\ 10000001 \\ 00010111 \\ 10110100 \\ 01010101 \\ 11000010 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= {\begin{bmatrix} 01100001 \\ 00100110 \\ 01110010 \\ 11001011 \\ 01011101 \\ 11101001 \\ 10101110 \\ 00000010 \end{bmatrix} = {A_{5}.}}} \end{matrix}$

Encryption at the 10^(th) Diffusion-Center (1,8):

$\begin{matrix} {A_{9}^{1} = {{A_{9}{F\left( {1,8} \right)}} = {A_{9} \oplus {A_{9}x_{1}} \oplus {A_{9}y_{8}} \oplus S}}} \\ {= {\begin{bmatrix} 01110011 \\ 10000110 \\ 10011100 \\ 10101100 \\ 01000101 \\ 10001011 \\ 00110101 \\ 10101001 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 01110011 \\ 10000110 \\ 10011100 \\ 10101100 \\ 01000101 \\ 10001011 \\ 00110101 \end{bmatrix} \oplus \begin{bmatrix} 11100110 \\ 00001100 \\ 00111000 \\ 01011000 \\ 10001010 \\ 00010110 \\ 01101010 \\ 01010010 \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= {\begin{bmatrix} 10010100 \\ 11111001 \\ 00100011 \\ 01101001 \\ 01100010 \\ 11011000 \\ 11010100 \\ 11001110 \end{bmatrix} = {A_{10}.}}} \end{matrix}$

Decryption at the 10^(th) Diffusion-Center (1,8):

$\begin{matrix} {A_{10}^{15} = {{A_{10}^{14}{F\left( {1,8} \right)}} = {A_{10}^{14} \oplus {A_{10}^{14}x_{1}} \oplus {A_{10}^{14}y_{8}} \oplus S}}} \\ {= {\begin{bmatrix} 00101110 \\ 10011000 \\ 00000011 \\ 10011010 \\ 01001010 \\ 10111111 \\ 10000110 \\ 11100101 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00101110 \\ 10011000 \\ 00000011 \\ 10011010 \\ 01001010 \\ 10111111 \\ 10000110 \end{bmatrix} \oplus \begin{bmatrix} 01011100 \\ 00110000 \\ 00000110 \\ 00110100 \\ 10010100 \\ 01111110 \\ 00001100 \\ 11001010 \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= {\begin{bmatrix} 01110011 \\ 10000110 \\ 10011100 \\ 10101100 \\ 01000101 \\ 10001011 \\ 00110101 \\ 10101001 \end{bmatrix} = {A_{9}.}}} \end{matrix}$

Decryption at the 5^(th) Diffusion-Center (4,5):

$\begin{matrix} {A_{5}^{15} = {{A_{5}^{14}{F\left( {4,5} \right)}} = {A_{5}^{14} \oplus {A_{5}^{14}x_{4}} \oplus {A_{5}^{14}y_{5}} \oplus S}}} \\ {= {\begin{bmatrix} 00011010 \\ 11111000 \\ 00011001 \\ 00111100 \\ 00010110 \\ 11000111 \\ 00100110 \\ 00111001 \end{bmatrix} \oplus \begin{bmatrix} 11111000 \\ 00011001 \\ 00111100 \\ 00000000 \\ 00111100 \\ 00010110 \\ 11000111 \\ 00100110 \end{bmatrix} \oplus \begin{bmatrix} 00110101 \\ 11110100 \\ 00110100 \\ 01110110 \\ 00100011 \\ 10000011 \\ 01000011 \\ 01110100 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= {\begin{bmatrix} 11010111 \\ 00010101 \\ 00010001 \\ 01000011 \\ 00001111 \\ 01011001 \\ 10101011 \\ 01100101 \end{bmatrix} = {A_{4}.}}} \end{matrix}$

Decryption at the 1^(st) Diffusion-Center (3,7):

$\begin{matrix} {A_{1}^{15} = {{A_{1}^{14}{F\left( {3,7} \right)}} = {A_{1}^{14} \oplus {A_{1}^{14}x_{3}} \oplus {A_{1}^{14}y_{7}} \oplus S}}} \\ {= {\begin{bmatrix} 11010110 \\ 10001001 \\ 00101000 \\ 00110101 \\ 01101011 \\ 01110011 \\ 01111010 \\ 11010111 \end{bmatrix} \oplus \begin{bmatrix} 10001001 \\ 00101000 \\ 00000000 \\ 00101000 \\ 00110101 \\ 01101011 \\ 01110011 \\ 01111010 \end{bmatrix} \oplus \begin{bmatrix} 10101101 \\ 00010000 \\ 01010000 \\ 01101000 \\ 11010101 \\ 11100101 \\ 11110101 \\ 10101101 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= {\begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix} = {A.}}} \end{matrix}$

EXAMPLE 2 The Function of Block-Diffusion in 2D

Supposes that

${S_{5 \times 5} = \begin{bmatrix} 10011 \\ 01101 \\ 10111 \\ 10010 \\ 11101 \end{bmatrix}},{\overset{.}{S} = \left( {1,1} \right)},{B_{2 \times 2} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}},{{\overset{.}{B} = \left( {1,1} \right)};}$ reads every diffusion-center in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle T=2²⁺¹=8, since d_(i)/b_(i)=4=2², and if 1 time on encryption, then 7 times on decryption. In math, inputs the plaintext A, then runs Â¹, Â₁ ¹, . . . Â₉ ¹, and outputs A₁, A₂, . . . A₁₀ during encryption; inputs the ciphertext A₁₀, then runs Â₁₀ ⁷, Â₉ ⁷, . . . Â₁ ⁷ and outputs A₉, . . . , A₁, A during decryption. The details on the order 1, 5, 10 are shown as below,

Â_(d̂_(i))(0) marked in boldface. Encryption at the 1^(st) Diffusion-Center (3,7):

$\begin{matrix} {{\hat{A}}^{1} = {{\hat{A}{F\left( {2,4} \right)}} =}} \\ {= {\hat{A} \oplus {\hat{A}{\hat{x}}_{2}} \oplus {\hat{A}{\hat{y}}_{4}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {3 - 1} \right)/2} \right\rceil + 1}},{\hat{p}}_{y}} \right.}}} \\ \left. {= {\left\lceil {\left( {7 - 1} \right)/2} \right\rceil + 1}} \right) \\ {= {\begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 01111010 \\ 01110100 \\ 00000000 \\ 00000000 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \end{bmatrix} \oplus \begin{bmatrix} 11001000 \\ 11000100 \\ 11101000 \\ 11010000 \\ 00100100 \\ 11111100 \\ 11111100 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= {\begin{bmatrix} 01000000 \\ 00000001 \\ 10010000 \\ 10100101 \\ 11010101 \\ 01110101 \\ 10001001 \\ 11111111 \end{bmatrix} = {A_{1}.}}} \end{matrix}$ Encryption at the 5^(th) Diffusion-Center (4,5):

$\begin{matrix} {{\hat{A}}_{4}^{1} = {{\hat{A}}_{4}{F\left( {3,3} \right)}}} \\ {= {{\hat{A}}_{4} \oplus {{\hat{A}}_{4}{\hat{x}}_{3}} \oplus {{\hat{A}}_{4}{\hat{y}}_{3}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {4 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {5 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 11000011 \\ 10100110 \\ 10001001 \\ 01000110 \\ 00110011 \\ 01100010 \\ 11011111 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 10001001 \\ 01000110 \\ 00110011 \\ 00000000 \\ 00000000 \\ 01000110 \\ 00110011 \\ 01100010 \end{bmatrix} \oplus \begin{bmatrix} {00000000} \\ {10010001} \\ {00100010} \\ {00010001} \\ {11000000} \\ {10000000} \\ {01110011} \\ {00000000} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= \begin{bmatrix} 01001010 \\ 01110001 \\ 10011000 \\ 01011110 \\ 11110101 \\ 10101111 \\ 10010110 \\ 01101100 \end{bmatrix}} \\ {= {A_{5}.}} \end{matrix}$ Encryption at the 10^(th) Diffusion-Center (1,8): (□, Zero in A_(ŷ), (5), 2^(nd) Col.)

$\begin{matrix} {{\hat{A}}_{9}^{1} = {{\hat{A}}_{9}{F\left( {1,5} \right)}}} \\ {= {{\hat{A}}_{9} \oplus {{\hat{A}}_{9}{\hat{x}}_{1}} \oplus {{\hat{A}}_{9}{\hat{y}}_{5}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {1 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {8 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 10011001 \\ 11110100 \\ 10001001 \\ 10001000 \\ 11011000 \\ 10000001 \\ 01110101 \\ 10011001 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 10011001 \\ 11110100 \\ 10001000 \\ 00010011 \\ 11011000 \\ 10000001 \end{bmatrix} \oplus \begin{bmatrix} {011001{\bullet 0}} \\ {110100{\bullet 0}} \\ {001000{\bullet 0}} \\ {010011{\bullet 0}} \\ {011000{\bullet 0}} \\ {000001{\bullet 0}} \\ {110101{\bullet 0}} \\ {011001{\bullet 0}} \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 11111100 \\ 00100100 \\ 00110000 \\ 10101010 \\ 00110001 \\ 10010110 \\ 01111001 \\ 01111100 \end{bmatrix}} \\ {= {A_{10}.}} \end{matrix}$ Decryption at the 10^(th) Diffusion-Center (1,8):

$\begin{matrix} {{\hat{A}}_{10}^{7} = {{\hat{A}}_{10}^{6}{F\left( {1,5} \right)}}} \\ {= {{\hat{A}}_{10}^{6} \oplus {{\hat{A}}_{10}^{6}{\hat{x}}_{1}} \oplus {{\hat{A}}_{10}^{6}{\hat{y}}_{5}} \oplus}} \\ {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {1 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {8 - 1} \right)/2} \right\rceil + 1}}} \right)} \\ {= {\begin{bmatrix} 01111000 \\ 01100100 \\ 01100101 \\ 01001110 \\ 10001100 \\ 11000011 \\ 11001101 \\ 00010010 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 01111000 \\ 01100100 \\ 01100101 \\ 01001110 \\ 10001100 \\ 11000011 \end{bmatrix} \oplus \begin{bmatrix} {111000{\bullet 0}} \\ {100100{\bullet 0}} \\ {100101{\bullet 0}} \\ {001110{\bullet 0}} \\ {001100{\bullet 0}} \\ {000011{\bullet 0}} \\ {001101{\bullet 0}} \\ {010010{\bullet 0}} \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 10011001 \\ 11110100 \\ 10001000 \\ 00010011 \\ 11011000 \\ 10000001 \\ 01110101 \\ 10011001 \end{bmatrix}} \\ {= {A_{9}.}} \end{matrix}$ Decryption at the 5^(th) diffusion-center (4,5):

$\begin{matrix} {{\hat{A}}_{5}^{7} = {{\hat{A}}_{5}^{6}{F\left( {3,3} \right)}}} \\ {= {{\hat{A}}_{5}^{6} \oplus {{\hat{A}}_{5}^{6}{\hat{x}}_{3}} \oplus {{\hat{A}}_{5}^{6}{\hat{y}}_{3}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {4 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {5 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 00101110 \\ 11111000 \\ 01011110 \\ 10111100 \\ 10100100 \\ 11000100 \\ 10110010 \\ 01101000 \end{bmatrix} \oplus \begin{bmatrix} 01011110 \\ 10111100 \\ 10100100 \\ 00000000 \\ 00000000 \\ 10111100 \\ 10100100 \\ 11000100 \end{bmatrix} \oplus \begin{bmatrix} {10110011} \\ {11100010} \\ {01110011} \\ {11110011} \\ {10010001} \\ {00010001} \\ {11000000} \\ {10100010} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= \begin{bmatrix} 11000011 \\ 10100110 \\ 10001001 \\ 01000110 \\ 00110011 \\ 01100010 \\ 11011111 \\ 00000000 \end{bmatrix}} \\ {= {A_{4}.}} \end{matrix}$ Decryption at the 1^(st) Diffusion-Center (3,7):

$\begin{matrix} {{\hat{A}}_{1}^{7} = {{\hat{A}}_{1}^{6}{F\left( {2,4} \right)}}} \\ {= {{\hat{A}}_{1}^{6} \oplus {{\hat{A}}_{1}^{6}{\hat{x}}_{2}} \oplus {{\hat{A}}_{1}^{6}{\hat{y}}_{4}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {3 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {7 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 01100010 \\ 00000000 \\ 00011000 \\ 10110001 \\ 00101111 \\ 10111100 \\ 01101111 \\ 10001100 \end{bmatrix} \oplus \begin{bmatrix} 00011000 \\ 10110001 \\ 00000000 \\ 00000000 \\ 00011000 \\ 10110001 \\ 00101111 \\ 10111100 \end{bmatrix} \oplus \begin{bmatrix} {10001000} \\ {00000000} \\ {01100000} \\ {11000100} \\ {10111100} \\ {11110000} \\ {10111100} \\ {00110000} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix}} \\ {= {A.}} \end{matrix}$

EXAMPLE 3 The Functions of Point-Diffusion and Block-Diffusion in 2D

Supposes

${S_{5 \times 5} = \begin{bmatrix} 10011 \\ 01101 \\ 10111 \\ 10010 \\ 11101 \end{bmatrix}},{\overset{.}{S} = \left( {1,1} \right)},{B_{2 \times 2} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}},{{\overset{.}{B} = \left( {1,1} \right)};}$ selects a switch set Y=[1011011101]; reads every diffusion-center and Y element in order, if from 1 to 10 on encryption, then from 10 back to 1 on decryption; counts the diffusion-cycle, if Y element is 1, then T=2³⁺¹=16 with point-diffusion, otherwise, T=2²⁺¹=8 with block-diffusion, and if 1 time on encryption, then 15 or 7 times on decryption.

In math, inputs the plaintext A, then runs A¹, Â₁ ¹, A₂ ¹, A₃ ¹, Â₄ ¹, A₅ ¹, A₆ ¹, A₇ ¹, Â₈ ¹, A₉ ¹ and outputs A₁, A₂, . . . , A₉, A₁₀ during encryption; inputs the ciphertext A₁₀, then runs A₁₀ ¹⁵, Â₉ ⁷, A₈ ¹⁵, A₇ ¹⁵, A₆ ¹⁵, Â₅ ⁷, A₄ ¹⁵, A₃ ¹⁵, Â₂ ⁷, A₁ ¹⁵ and outputs A₉, A₈, . . . , A₁, A during decryption. The details on the order 1, 5, 10 are shown as below, A_(d) _(i) (0) and

Â_(d̂_(i))(0) marked in boldface. Encryption at the 1^(st) Diffusion-Center (3,7): Y(1)=1, Point-Diffusion.

$\begin{matrix} {A^{1} = {{AF}\left( {3,7} \right)}} \\ {= {A \oplus {Ax}_{3} \oplus {Ay}_{7} \oplus S}} \\ {= {\begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix} \oplus \begin{bmatrix} 10110001 \\ 01111010 \\ 00000000 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \end{bmatrix} \oplus \begin{bmatrix} {11100101} \\ {01100000} \\ {11110101} \\ {11101000} \\ {00010000} \\ {11111101} \\ {11111101} \\ {00000000} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 10100110 \\ 10101011 \\ 10001101 \\ 11100111 \\ 11101111 \\ 10001001 \\ 11111110 \\ 11111111 \end{bmatrix}} \\ {= A_{1}} \end{matrix}$ Encryption at the 5^(th) Diffusion-Center (4,5): Y(5)=0, Block-Diffusion.

$\begin{matrix} {{\hat{A}}_{4}^{1} = {{\hat{A}}_{4}{F\left( {3,3} \right)}}} \\ {= {{\hat{A}}_{4} \oplus {A_{4}{\hat{x}}_{3}} \oplus {{\hat{A}}_{4}{\hat{y}}_{3}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {4 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {5 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 11001000 \\ 00011010 \\ 10000111 \\ 11010010 \\ 01000111 \\ 11100010 \\ 11010101 \\ 00010110 \end{bmatrix} \oplus \begin{bmatrix} 10000111 \\ 11010010 \\ 01000111 \\ 00000000 \\ 00000000 \\ 11010010 \\ 01000111 \\ 11100010 \end{bmatrix} \oplus \begin{bmatrix} {00100010} \\ {01100010} \\ {00010001} \\ {01000000} \\ {00010001} \\ {10000000} \\ {01010001} \\ {01010001} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= \begin{bmatrix} 01101101 \\ 10101010 \\ 11010001 \\ 10011011 \\ 01010000 \\ 10111011 \\ 11001010 \\ 10101011 \end{bmatrix}} \\ {= A_{5}} \end{matrix}$ Encryption at the 10^(th) Diffusion-Center (1,8): Y(10)=1, Point-Diffusion.

$\begin{matrix} {A_{9}^{1} = {A_{9}{F\left( {1,8} \right)}}} \\ {= {A_{9} \oplus {A_{9}x_{1}} \oplus {A_{9}y_{8}} \oplus S}} \\ {= {\begin{bmatrix} 00110000 \\ 11000111 \\ 00001010 \\ 10000100 \\ 00101100 \\ 11110100 \\ 00000111 \\ 10011011 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00110000 \\ 11000111 \\ 00001010 \\ 10000100 \\ 00101100 \\ 11110100 \\ 00000111 \end{bmatrix} \oplus \begin{bmatrix} {01100000} \\ {10001110} \\ {00010100} \\ {00001000} \\ {01011000} \\ {11101000} \\ {00001110} \\ {00110110} \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 01010001 \\ 01111001 \\ 11011000 \\ 10000111 \\ 11110001 \\ 00110000 \\ 11111101 \\ 10101010 \end{bmatrix}} \\ {= A_{10}} \end{matrix}$ Decryption at the 10^(th) Diffusion-Center (1,8): Y(10)=1, Point-Diffusion.

$\begin{matrix} {A_{10}^{15} = {A_{10}^{14}{F\left( {1,8} \right)}}} \\ {= {A_{10}^{14} \oplus {A_{10}^{14}x_{1}} \oplus {A_{10}^{14}y_{8}} \oplus S}} \\ {= {\begin{bmatrix} 11101111 \\ 00011000 \\ 11110001 \\ 00101100 \\ 11111111 \\ 11111001 \\ 10101010 \\ 11101111 \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 11101111 \\ 00011000 \\ 11110001 \\ 00101100 \\ 11111111 \\ 11111001 \\ 10101010 \end{bmatrix} \oplus \begin{bmatrix} {11011110} \\ {00110000} \\ {11100010} \\ {01011000} \\ {11111110} \\ {11110010} \\ {01010100} \\ {11011110} \end{bmatrix} \oplus \begin{bmatrix} 00000001 \\ 00000000 \\ 00000001 \\ 00000001 \\ 00000001 \\ 00000000 \\ 00000000 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 00110000 \\ 11000111 \\ 00001010 \\ 10000100 \\ 00101100 \\ 11110100 \\ 00000111 \\ 10011011 \end{bmatrix}} \\ {= A_{9}} \end{matrix}$ Decryption at the 5^(th) Diffusion-Center (4,5): Y(5)=0, Block-Diffusion.

$\begin{matrix} {{\hat{A}}_{5}^{7} = {{\hat{A}}_{5}^{6}{F\left( {3,3} \right)}}} \\ {= {{\hat{A}}_{5}^{6} \oplus {{\hat{A}}_{5}^{6}{\hat{x}}_{3}} \oplus {{\hat{A}}_{5}^{6}{\hat{y}}_{3}} \oplus {S\left( {{{\because{\hat{p}}_{x}} = {\left\lceil {\left( {4 - 1} \right)/2} \right\rceil + 1}},{{\hat{p}}_{y} = {\left\lceil {\left( {5 - 1} \right)/2} \right\rceil + 1}}} \right)}}} \\ {= {\begin{bmatrix} 11101100 \\ 10100011 \\ 10010111 \\ 00111001 \\ 01000111 \\ 10010000 \\ 00101110 \\ 00101010 \end{bmatrix} \oplus \begin{bmatrix} 10010111 \\ 00111001 \\ 01000001 \\ 00000000 \\ 00000000 \\ 00111001 \\ 01000001 \\ 10010000 \end{bmatrix} \oplus \begin{bmatrix} {10110011} \\ {10000000} \\ {01010001} \\ {11100010} \\ {00000000} \\ {01000000} \\ {10110011} \\ {10100010} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00001001 \\ 00000110 \\ 00001011 \\ 00001001 \\ 00001110 \end{bmatrix}}} \\ {= \begin{bmatrix} 11001000 \\ 00011010 \\ 10000111 \\ 11010010 \\ 01000111 \\ 11100010 \\ 11010101 \\ 00010110 \end{bmatrix}} \\ {= A_{4}} \end{matrix}$ Decryption at the 1^(st) Diffusion-Center (3,7): Y(1)=1, Point-Diffusion.

$\begin{matrix} {A_{1}^{15} = {A_{1}^{14}{F\left( {3,7} \right)}}} \\ {= {A_{1}^{14} \oplus {A_{1}^{14}x_{3}} \oplus {A_{1}^{14}y_{7}} \oplus S}} \\ {= {\begin{bmatrix} 11010110 \\ 10001001 \\ 00101000 \\ 00110101 \\ 01101011 \\ 01110011 \\ 01111010 \\ 11010111 \end{bmatrix} \oplus \begin{bmatrix} 10001001 \\ 00101000 \\ 00000000 \\ 00101000 \\ 00110101 \\ 01101011 \\ 01110011 \\ 01111010 \end{bmatrix} \oplus \begin{bmatrix} {10101101} \\ {00010000} \\ {01010000} \\ {01101000} \\ {11010101} \\ {11100101} \\ {11110101} \\ {10101101} \end{bmatrix} \oplus \begin{bmatrix} 00000000 \\ 00000000 \\ 00000010 \\ 00000001 \\ 00000010 \\ 00000010 \\ 00000011 \\ 00000000 \end{bmatrix}}} \\ {= \begin{bmatrix} 11110010 \\ 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 11111111 \\ 00000000 \end{bmatrix}} \\ {= A} \end{matrix}$

EXAMPLE 4 The Function of Point-Diffusion in 3D

Supposes a plaintext A: let Table 1-1 overlap for 8 times to shape a 8×8×8 binary matrix, shown as a 8×8 matrix in ASCII code format as in Table 2-1. To figure out the later 3D calculation clearly with Table 2-1, the row stands for a x-y plane, namely Table 1-1, and all rows resolve as the axis z.

TABLE 2-1 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72 73 6d 6f 6f 74 68 65 72

Suppose a password: “YourlipsY”, its ASCII code is 59 6f 75 72 6c 69 70 73 59. For applying to the plaintext, the ASCII code: first, subtracts 8 if a digit >7 and leaves 51 67 75 72 64 61 70 73 51; second, every three-digit forms a division; third, adds 1 to each digit; Table 2-2 shows that the password includes 6 diffusion-centers.

TABLE 2-2 ASCII 516 775 726 461 707 351 First Dimension 6 8 8 5 8 4 Second Dimension 2 8 3 7 1 6 Third Dimension 7 6 7 2 8 2

Supposes S_(1×1×1)=1, {dot over (S)}=(1,1,1); reads every diffusion-center in order, if from 1 to 6 on encryption, then from 6 back to 1 on decryption; counts the diffusion-cycle T=2³⁺¹=16, if 1 time on encryption, then 15 times on decryption. In math, inputs the plaintext A, then runs A¹, A₁ ¹, . . . A₅ ¹ and outputs A₁, A₂, . . . A₆ during encryption; inputs the ciphertext A₆, then runs A₆ ¹⁵, A₅ ¹⁵, . . . A₁ ¹⁵ and outputs A₅, . . . , A₁, A during decryption. The details on the order 1, 6 are shown as below, A_(d) _(i) (0) marked in boldface.

Encryption at the 1^(st) Diffusion-Center (6,2,7): A ¹ =AF(6,2,7)=A⊕Ax ₆ ⊕Ay ₂ ⊕Az ₇ ⊕S=A ₁;

Considering that the row of Table 2-1 means a x-y plane, it can be figured out by the 3D scheme through rearranging every plane then placing to the corresponding row of 2D table, as Ax₆ and Ay₂ as follows.

${{a\mspace{14mu}{plane}\mspace{14mu}{of}\mspace{14mu}{Ax}_{6}} = \begin{bmatrix} 10110001 \\ 01111010 \\ 01110100 \\ 10001001 \\ 11111111 \\ 00000000 \\ 11111111 \\ 11111111 \end{bmatrix}},{{{Ax}_{6} = \begin{bmatrix} {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \\ {d\; 9} & {d\; 6} & {d\; 7} & {d\; 7} & {da} & {d\; 4} & {d\; 2} & {d\; 9} \end{bmatrix}};}$ ${{a\mspace{14mu}{plane}\mspace{14mu}{of}\mspace{14mu}{Ay}_{2}} = \begin{bmatrix} 10111001 \\ 00011000 \\ 10111101 \\ 10111010 \\ 00000000 \\ 10111111 \\ 10111111 \\ 00000000 \end{bmatrix}},{{{Ay}_{2} = \begin{bmatrix} {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \\ {6\; d} & 00 & {6\; d} & {6\; f} & {6\; f} & 74 & 68 & 65 \end{bmatrix}};}$

In addition, S is anchored to position (6,2,7), see below, value 1 found at p_(x)=6, p_(y)=2 on the 7^(th) plane (p_(z)=7).

${{{the}\mspace{14mu} 7^{th}\mspace{14mu}{plane}\mspace{14mu}{of}\mspace{14mu} S} = \begin{bmatrix} 00000000 \\ 00000000 \\ 00000000 \\ 00000000 \\ 00000000 \\ 01000000 \\ 00000000 \\ 00000000 \end{bmatrix}},{S = {\begin{bmatrix} 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 20 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \end{bmatrix}.}}$ Encryption at the 6^(th) Diffusion-Center (4,6,2):

A₅F(4, 6, 2) = A₅ ⊕ A₅x₄ ⊕ A₅y₆ ⊕ A₅z₂ ⊕ S = A₆; ${A_{5} = \begin{bmatrix} {c\; 4} & 11 & {3a} & {b\; 7} & {7a} & 64 & 01 & {ed} \\ {8f} & {c\; 0} & {8e} & {8d} & {a\; 1} & {b\; 7} & {cb} & 00 \\ 53 & {9c} & 48 & 26 & {ee} & {eb} & {8b} & 23 \\ {5b} & {5b} & 27 & 47 & 91 & {b\; 1} & {5c} & {cb} \\ {b\; 8} & 34 & 67 & {4a} & {9f} & {b\; 3} & 74 & {4c} \\ 92 & 17 & {5d} & {5a} & {7e} & {7d} & 84 & {0f} \\ {a\; 0} & 19 & {5d} & {e\; 6} & 48 & {7a} & 82 & {c\; 3} \\ {bf} & {ac} & 38 & {df} & 13 & {2f} & {d\; 8} & {1a} \end{bmatrix}};$ ${{A_{5}z_{2}} = \begin{bmatrix} {8f} & {c\; 0} & {8e} & {8d} & {a\; 1} & {b\; 7} & {cb} & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ {8f} & {c\; 0} & {8e} & {8d} & {a\; 1} & {b\; 7} & {cb} & 00 \\ 53 & {9c} & 48 & 26 & {ee} & {eb} & {8b} & 23 \\ {5b} & {5b} & 27 & 47 & 91 & {b\; 1} & {5c} & {cb} \\ {b\; 8} & 34 & 67 & {4a} & {9f} & {b\; 3} & 74 & {4c} \\ 92 & 17 & {5d} & {5a} & {7e} & {7d} & 84 & {0f} \\ {a\; 0} & 19 & {5d} & {e\; 6} & 48 & {7a} & 82 & {c\; 3} \end{bmatrix}};$ ${{A_{5}x_{4}} = \begin{bmatrix} 82 & 20 & 75 & 63 & {f\; 5} & {c\; 2} & 00 & {d\; 6} \\ 17 & 80 & 17 & 16 & 40 & 63 & 95 & 00 \\ {a\; 1} & 36 & 94 & 43 & {d\; 7} & {d\; 5} & 15 & 41 \\ {b\; 5} & {b\; 5} & 43 & 83 & 20 & 60 & {b\; 6} & 95 \\ 74 & 62 & {c\; 3} & 95 & 37 & 61 & {e\; 2} & 96 \\ 21 & 23 & {b\; 6} & {b\; 5} & {f\; 7} & {f\; 6} & 02 & 17 \\ 40 & 34 & {b\; 6} & {c\; 3} & 94 & {f\; 5} & 01 & 81 \\ 77 & 56 & 74 & {b\; 7} & 21 & 57 & {b\; 4} & 35 \end{bmatrix}};$ ${S = \begin{bmatrix} 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 08 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \end{bmatrix}};$ ${{A_{5}y_{6}} = \begin{bmatrix} 11 & {3a} & {b\; 7} & {7a} & 64 & 00 & 64 & 01 \\ {c\; 0} & {8e} & {8d} & {a\; 1} & {b\; 7} & 00 & {b\; 7} & {cb} \\ {9c} & 48 & 26 & {ee} & {eb} & 00 & {eb} & {8b} \\ {5b} & 27 & 47 & 91 & {b\; 1} & 00 & {b\; 1} & {5c} \\ 34 & 67 & {4a} & {9f} & {b\; 3} & 00 & {b\; 3} & 74 \\ 17 & {5d} & {5a} & {7e} & {7d} & 00 & {7d} & 84 \\ 19 & {5d} & {e\; 6} & 48 & {7a} & 00 & {7a} & 82 \\ {ac} & 38 & {df} & 13 & {2f} & 00 & {2f} & {d\; 8} \end{bmatrix}};$ $A_{6} = {\begin{bmatrix} {d\; 8} & {cb} & 76 & 23 & {4a} & 11 & {ae} & {3a} \\ 58 & {ce} & 14 & {3a} & 56 & {dc} & {e\; 9} & {cb} \\ {e\; 1} & 22 & 74 & 06 & 73 & 89 & {be} & {e\; 9} \\ {e\; 6} & 55 & {6b} & 73 & {ee} & {3a} & {d\; 0} & 21 \\ {a\; 3} & {6a} & {c\; 9} & 07 & {8a} & 63 & 79 & 65 \\ {1c} & {5d} & {d\; 6} & {db} & {6b} & 38 & {8f} & {d\; 0} \\ {6b} & 67 & 50 & 37 & {d\; 8} & {f\; 2} & {7d} & {cf} \\ {c\; 4} & {db} & {ce} & {9d} & 55 & 02 & {c\; 1} & 34 \end{bmatrix}.}$ Decryption at the 6^(th) Diffusion-Center (4,6,2):

A₆¹⁵ = A₆¹⁴F(4, 6, 2) = A₆¹⁴ ⊕ A₆¹⁴x₄ ⊕ A₆¹⁴y₆ ⊕ A₆¹⁴z₂ ⊕ S = A₅; ${A_{6}^{14} = \begin{bmatrix} 24 & 31 & {7d} & {3a} & {e\; 8} & {fb} & {8d} & {c\; 1} \\ 93 & {3d} & {8b} & 10 & {a\; 1} & {6a} & 61 & 21 \\ {a\; 6} & 25 & {c\; 6} & 86 & {ee} & 81 & {d\; 0} & {4a} \\ 47 & 39 & 33 & {b\; 3} & 91 & 10 & 71 & 50 \\ {\; 50} & {0f} & 15 & 63 & {9f} & 62 & 25 & {ee} \\ {c\; 6} & 87 & {9c} & {e\; 2} & {7e} & {0a} & {9d} & 28 \\ {9f} & {ce} & {c\; 7} & 85 & 48 & {d\; 0} & {ba} & 31 \\ {3c} & {6a} & {dd} & 94 & 13 & {aa} & {bf} & 77 \end{bmatrix}};$ ${{A_{6}^{14}x_{4}} = \begin{bmatrix} 42 & 60 & {f\; 6} & 75 & {d\; 4} & {f\; 5} & 16 & 80 \\ 21 & 76 & 15 & 20 & 76 & {d\; 5} & {c\; 0} & 40 \\ 43 & 42 & 83 & 03 & 61 & 00 & {b\; 6} & 95 \\ 83 & 74 & 61 & 61 & 21 & 20 & {e\; 0} & {a\; 0} \\ {a\; 0} & 17 & 22 & {c\; 1} & {b\; 5} & {c\; 1} & 42 & {d\; 7} \\ 83 & 03 & 36 & {c\; 1} & 35 & 15 & 36 & 54 \\ 37 & 97 & 83 & 02 & 01 & {a\; 0} & 75 & 60 \\ 76 & {d\; 5} & {b\; 6} & 22 & {d\; 6} & 55 & 77 & {e\; 3} \end{bmatrix}};$ ${{A_{6}^{14}y_{6}} = \begin{bmatrix} 31 & {7\; d} & {3a} & {e\; 8} & {fb} & 00 & {fb} & {8d} \\ {3d} & {8b} & 10 & {bd} & {6a} & 00 & {6a} & 61 \\ 25 & {c\; 6} & 86 & {b\; 3} & 81 & 00 & 81 & {dd} \\ 39 & 33 & {b\; 3} & 13 & 10 & 00 & 10 & 71 \\ {0f} & 15 & 63 & {5b} & 62 & 00 & 62 & 25 \\ 87 & {9c} & {e\; 2} & {1a} & {0a} & 00 & {0a} & {9d} \\ {ce} & {c\; 7} & 85 & 83 & {d\; 0} & 00 & {d\; 0} & {ba} \\ {6a} & {dd} & 94 & {ec} & {aa} & 00 & {aa} & {bf} \end{bmatrix}};$ ${{A_{6}^{14}z_{2}} = \begin{bmatrix} 93 & {3d} & {8b} & 10 & {bd} & {6a} & 61 & 21 \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ 93 & {3d} & {8b} & 10 & {bd} & {6a} & 61 & 21 \\ {a\; 6} & 25 & {c\; 6} & 86 & {b\; 3} & 81 & {dd} & {4a} \\ 47 & 39 & 33 & {b\; 3} & 13 & 10 & 71 & 50 \\ 50 & {0f} & 15 & 63 & {5b} & 62 & 25 & {ee} \\ {c\; 6} & 87 & {9c} & {e\; 2} & {1a} & {0a} & {9d} & 28 \\ {9f} & {ce} & {c\; 7} & 85 & 83 & {d\; 0} & {ba} & 31 \end{bmatrix}};$ Decryption at the 1^(st) Diffusion-Center (6,2,7):

${A_{1}^{15} = {{A_{1}^{14}{F\left( {6,2,7} \right)}} = {{A_{1}^{14} \oplus {A_{1}^{14}x_{6}} \oplus {A_{1}^{14}y_{2}} \oplus {A_{1}^{14}z_{7}} \oplus S} = {{A.A_{1}^{14}} = \begin{bmatrix} 53 & {1d} & {4f} & {6e} & 77 & {6e} & {e\; 8} & 28 \\ {b\; 4} & {7e} & {ba} & 99 & {b\; 6} & {a\; 6} & 37 & {e\; 6} \\ {e\; 3} & {cc} & {f\; 6} & 40 & 00 & {2b} & {af} & 70 \\ {8d} & 27 & {8e} & 59 & 34 & 16 & {b\; 6} & 75 \\ {df} & 19 & {c\; 1} & 34 & 43 & 41 & {9e} & {6b} \\ {3a} & 78 & 37 & {c\; 0} & 82 & {ea} & {b\; 5} & 52 \\ {6c} & {c\; 9} & {7b} & 18 & 37 & {d\; 5} & 60 & {1c} \\ {3a} & 78 & 37 & {c\; 0} & 82 & {ea} & {b\; 5} & 52 \end{bmatrix}}}}};$ ${{A_{1}^{14}x_{6}} = \begin{bmatrix} 89 & {0e} & 87 & {d\; 7} & {db} & {d\; 7} & {d\; 4} & 54 \\ {5a} & {df} & {5d} & {0c} & {5b} & 53 & {5b} & {d\; 3} \\ {d\; 1} & 86 & {db} & 80 & 00 & 55 & 57 & {d\; 8} \\ 06 & 53 & 07 & {8c} & {5a} & {0b} & {5b} & {da} \\ {8f} & {0c} & 80 & {5a} & 81 & 80 & {0f} & {d\; 5} \\ {5d} & {dc} & {5b} & 80 & 01 & {d\; 5} & {5a} & 89 \\ {d\; 6} & 84 & {dd} & {0c} & {5b} & {8a} & {d\; 0} & {0e} \\ {5d} & {dc} & {5b} & 80 & 01 & {d\; 5} & {5a} & 89 \end{bmatrix}};$ ${{A_{1}^{14}y_{2}} = \begin{bmatrix} {1d} & 00 & {1d} & {4f} & {6e} & 77 & {6e} & {e\; 8} \\ {7e} & 00 & {7e} & {ba} & 99 & {b\; 6} & {a\; 6} & 37 \\ {cc} & 00 & {cc} & {f\; 6} & 40 & 00 & {2b} & {af} \\ 27 & 00 & 27 & {8e} & 59 & 34 & 16 & {b\; 6} \\ 19 & 00 & 19 & {c\; 1} & 34 & 43 & 41 & {9e} \\ 78 & 00 & 78 & 37 & {c\; 0} & 82 & {ea} & {b\; 5} \\ {c\; 9} & 00 & {c\; 9} & {7b} & 18 & 37 & {d\; 5} & 60 \\ 78 & 00 & 78 & {\; 37} & {c\; 0} & 82 & {ea} & {b\; 5} \end{bmatrix}};$ ${{A_{1}^{14}z_{7}} = \begin{bmatrix} {b\; 4} & {7e} & {ba} & 99 & {b\; 6} & {a\; 6} & 37 & {e\; 6} \\ {e\; 3} & {cc} & {f\; 6} & 40 & 00 & {2b} & {af} & 70 \\ {8d} & 27 & {8e} & 59 & 34 & 16 & {b\; 6} & 75 \\ {df} & 19 & {c\; 1} & 34 & 43 & 41 & {9e} & {6b} \\ {3a} & 78 & 37 & {c\; 0} & 82 & {ea} & {b\; 5} & 52 \\ {6c} & {c\; 9} & {7b} & 18 & 37 & {d\; 5} & 60 & {1c} \\ 00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 \\ {6c} & {c\; 9} & {7b} & 18 & 37 & {d\; 5} & 60 & {1c} \end{bmatrix}};$

In summation of the above description, the present invention herein complies with the constitutional, statutory, regulatory and treaty, patent application requirements and is herewith submitted for patent application. However, the description and its accompanied drawings are used for describing preferred embodiments of the present invention, and it is to be understood that the invention is not limited thereto. To the contrary, it is intended to cover various modifications and similar arrangements and procedures, and the scope of the appended claims therefore should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements and procedures. 

1. A computer-implemented cryptographic method, the method comprising: a memory including a plaintext M run by at least one variable module, and a computer-processor configured to perform the steps of: selecting dimension of M, wherein M forms . . . n-dimension binary matrix; selecting a diffusion-center P, wherein P expresses . . . n-dimension position; selecting a medium S, wherein S is . . . n-dimension position; selecting a function of point-diffusion . . . ; and setting a diffusion-cycle T, wherein . . . the method further comprising steps of: (a) encrypting M, wherein . . . ; and (b) decrypting C, wherein . . . ; wherein the said encrypting and decrypting steps are run by the said computer-processor.
 2. The cryptographic method according to claim 1, wherein said function of point-diffusion comprises that: S overlaps A, {dot over (S)} anchoring to P; Ad_(ip), 1≦i≦n, expresses a series of n−1 dimensional binary matrixes A_(d) _(i) , Ad_(ip)=[A_(d) _(i) (2), . . . , A_(d) _(i) (p_(i)), A_(d) _(i) (0), A_(d) _(i) (p_(i)), . . . , A_(d) _(i) (d_(i)−1)], on d_(i) axis in order, wherein A_(d) _(i) (p_(i)) represents the original matrix at p_(i) position, and A_(d) _(i) (0), expresses a zero matrix filling at p_(i) position.
 3. The cryptographic method according to claim 1, further including a password, wherein said password has at least one variable data, said data read by said module, each including: (p₁, p₂, . . . , p_(n)) and/or d₁×d₂× . . . ×d_(n) and/or ({dot over (s)}₁, {dot over (s)}₂, . . . {dot over (s)}_(n)) and/or s₁×s₂× . . . ×s_(n) and/or S and/or T_(E) and/or T_(D).
 4. A computer-implemented cryptographic method, the method comprising: a memory including a plaintext M run by at least one variable module, and a computer-processor configured to perform the steps of: selecting dimension of M, wherein M forms . . . n-dimension binary matrix; selecting a diffusion-center P, wherein P expresses . . . n-dimension position; selecting a medium S, wherein S is . . . n-dimension position; selecting a block B, where . . . n-dimension position; selecting a function of block-diffusion . . . n-dimension position; and setting a diffusion-cycle T, wherein . . . the method further comprising steps of: (a) encrypting M, wherein . . . ; and (b) decrypting C, wherein . . . ; wherein the said encrypting and decrypting steps are run by the said computer-processor.
 5. The cryptographic method according to claim 4, wherein said function of block-diffusion comprises that: S overlaps A based on {dot over (S)} anchoring to P; Â expresses A by B unit based on {dot over (B)} anchoring to P, wherein Â is {circumflex over (d)}₁×{circumflex over (d)}₂× . . . ×{circumflex over (d)}_(n) n-dimension binary matrix which has a block diffusion-center {circumflex over (P)}, wherein {circumflex over (P)} expresses a ({circumflex over (p)}₁, {circumflex over (p)}₂, . . . {circumflex over (p)}_(n)) n-dimension position; lets {circumflex over (d)}_(i)=┌(p_(i)−{dot over (b)}_(i))/b_(i)┐+┌(d_(i)−p_(i)+{dot over (b)}_(i))/b_(i)┐, and {circumflex over (p)}_(i)=|(p_(i)−{dot over (b)}_(i))/b_(i)|+1, 1≦i≦n; Â{circumflex over (d)}_(i{circumflex over (p)}), 1≦i≦n, expresses a series of n−1 dimensional binary matrixes Â_({circumflex over (d)}) _(i) , Âd̂_(ip̂) = [Â_(d̂_(i))(2), …  , Â_(d̂_(i))(p̂_(i)), Â_(d̂_(i))(0), Â_(d̂_(i))(p̂_(i)), …  , Â_(d̂_(i))(d̂_(i) − 1)], on {circumflex over (d)}_(i) axis in order, wherein Â_(d̂_(i))(p̂_(i)) represents the original matrix at {circumflex over (p)}_(i) position, and Â_(d̂_(i))(0) expresses a zero matrix filling at {circumflex over (p)}_(i) position.
 6. The cryptographic method according to claim 4, further including a password, wherein said password has at least one variable data, said data read by said module, each including: (p₁, p₂, . . . , p_(n)) and/or d₁×d₂× . . . ×d_(n) and/or ({dot over (s)}₁, {dot over (s)}₂, . . . {dot over (s)}_(n)) and/or s₁×s₂× . . . ×s_(n) and/or S and/or ({dot over (b)}₁, {dot over (b)}₂, . . . , {dot over (b)}_(n)) and/or b₁×b₂× . . . ×b_(n) and/or T_(E) and/or T_(D).
 7. A computer-implemented cryptographic method, the method comprising: a memory including a plaintext M run by at least one variable module, and a computer-processor configured to perform the steps of: selecting dimension of M, wherein M forms . . . n-dimension binary matrix; selecting a diffusion-center P, wherein P expresses . . . n-dimension position; selecting a medium S, wherein S is . . . n-dimension position; selecting a block B, where . . . n-dimension position; selecting a switch Y, wherein Y represents . . . a function of block-diffusion; selecting said function of point-diffusion . . . ; setting a diffusion-cycle T1, wherein . . . ; setting a diffusion-cycle T2, wherein . . . ; and the method further comprising steps of: (a) encrypting M, wherein . . . ; and (b) decrypting C, wherein . . . ; wherein the said encrypting and decrypting steps are run by the said computer-processor.
 8. The cryptographic method according to claim 7, wherein said function of point-diffusion comprises that: S overlaps A, {dot over (S)} anchoring to P; Ad_(ip), 1≦i≦n, expresses a series of n−1 dimensional binary matrixes A_(d) _(i) , Ad_(ip)=[A_(d) _(i) (2), . . . , A_(d) _(i) (p_(i)), A_(d) _(i) (0), A_(d) _(i) (p_(i)), . . . , A_(d) _(i) (d_(i)−1)], on d_(i) axis in order, wherein A_(d) _(i) (p_(i)) represents the original matrix at p_(i) position, and A_(d) _(i) (0) expresses a zero matrix filling at p_(i) position.
 9. The cryptographic method according to claim 7, wherein said function of block-diffusion comprises that: S overlaps A based on {dot over (S)} anchoring to P; Â expresses A by B unit based on {dot over (B)} anchoring to P, wherein Â is {circumflex over (d)}₁×{circumflex over (d)}₂× . . . ×{circumflex over (d)}_(n) n-dimension binary matrix which has a block diffusion-center {circumflex over (P)}, wherein {circumflex over (P)} expresses a ({circumflex over (p)}₁, {circumflex over (p)}₂, . . . , {circumflex over (p)}_(n)) n-dimension position; lets {circumflex over (d)}_(i)=┌(p_(i)−{dot over (b)}_(i))/b_(i)┐+┌(d_(i)−p_(i)+{dot over (b)}_(i))/b_(i)┐, and {circumflex over (p)}_(i)=┌(p_(i)−{dot over (b)}_(i))/b_(i)┐+1, 1≦i≦n; Â{circumflex over (d)}_(i{circumflex over (p)}), 1≦i≦n, expresses a series of n−1 dimensional binary matrixes Â_({circumflex over (d)}) _(i) , Â_(d̂_(i)), Âd̂_(ip̂) = [Â_(d̂_(i))(2), …  , Â_(d̂_(i))(p̂_(i)), Â_(d̂_(i))(0), Â_(d̂_(i))(p̂_(i)), …  , Â_(d̂_(i))(d̂_(i) − 1)], on {circumflex over (d)}_(i) axis in order, wherein Â_(d̂_(i))(p̂_(i)) represents the original matrix at {circumflex over (p)}_(i) position, and Â_(d̂_(i))(0) expresses a zero matrix filling at {circumflex over (p)}_(i) position.
 10. The cryptographic method according to claim 7, further including a password, wherein said password has at least one variable data, said data read by said module, each including: (p₁, p₂, . . . p_(n)) and/or d₁×d₂× . . . ×d_(n) and/or ({dot over (s)}₁, {dot over (s)}₂, . . . {dot over (s)}_(n)) and/or s₁×s₂× . . . ×s_(n) and/or S and/or ({dot over (b)}₁, {dot over (b)}₂, . . . , {dot over (b)}_(n)) and/or b₁×b₂× . . . ×b_(n) and/or Y and/or T_(E1) and/or T_(D1) and/or T_(E2) and/or T_(D2). 